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A cone is a
three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...
geometric shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie ...
that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...
. A cone is formed by a set of
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
s,
half-line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segm ...
s, or lines connecting a common point, the apex, to all of the points on a base that is in a plane that does not contain the apex. Depending on the author, the base may be restricted to be a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
, any one-dimensional
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
in the plane, any closed one-dimensional figure, or any of the above plus all the enclosed points. If the enclosed points are included in the base, the cone is a
solid object In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces (i.e., 3D geometry). Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), in ...
; otherwise it is a
two-dimensional In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
object in three-dimensional space. In the case of a solid object, the boundary formed by these lines or partial lines is called the ''lateral surface''; if the lateral surface is unbounded, it is a
conical surface In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the ''apex'' or ''vertex'' — and any point of some fixed space curve — the ''di ...
. In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, it extends infinitely far. In the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a double cone on one side of the apex is called a ''nappe''. The axis of a cone is the straight line (if any), passing through the apex, about which the base (and the whole cone) has a
circular symmetry In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself. Rotational circular symmetry is isomorphic with the circle group in the complex plane, or th ...
. In common usage in elementary
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, cones are assumed to be right circular, where ''circular'' means that the base is a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
and ''right'' means that the axis passes through the centre of the base at right angles to its plane. If the cone is right circular the intersection of a plane with the lateral surface is a
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
. In general, however, the base may be any shapeGrünbaum, ''
Convex Polytopes ''Convex Polytopes'' is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, Micha Perl ...
'', second edition, p. 23.
and the apex may lie anywhere (though it is usually assumed that the base is bounded and therefore has finite
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
, and that the apex lies outside the plane of the base). Contrasted with right cones are oblique cones, in which the axis passes through the centre of the base non-perpendicularly. A cone with a
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
al base is called a
pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...
. Depending on the context, "cone" may also mean specifically a
convex cone In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every . W ...
or a
projective cone A projective cone (or just cone) in projective geometry is the union of all lines that intersect a projective subspace ''R'' (the apex of the cone) and an arbitrary subset ''A'' (the basis) of some other subspace ''S'', disjoint from ''R''. In ...
. Cones can also be generalized to
higher dimensions In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
.


Further terminology

The perimeter of the base of a cone is called the "directrix", and each of the line segments between the directrix and apex is a "generatrix" or "generating line" of the lateral surface. (For the connection between this sense of the term "directrix" and the directrix of a conic section, see
Dandelin spheres In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plan ...
.) The "base radius" of a circular cone is the
radius In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...
of its base; often this is simply called the radius of the cone. The
aperture In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture and focal length of an optical system determine the cone angle of a bundle of rays that come to a focus in the image plane. An ...
of a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes an angle ''θ'' to the axis, the aperture is 2''θ''. A cone with a region including its apex cut off by a plane is called a " truncated cone"; if the truncation plane is parallel to the cone's base, it is called a frustum. An "elliptical cone" is a cone with an elliptical base. A "generalized cone" is the surface created by the set of lines passing through a vertex and every point on a boundary (also see
visual hull A visual hull is a geometric entity created by shape-from-silhouette 3D reconstruction technique introduced by A. Laurentini. This technique assumes the foreground object in an image can be separated from the background. Under this assumption, ...
).


Measurements and equations


Volume

The
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
V of any conic solid is one third of the product of the area of the base A_B and the height h :V = \fracA_B h. In modern mathematics, this formula can easily be computed using calculus — it is, up to scaling, the integral \int x^2 dx = \tfrac x^3 Without using calculus, the formula can be proven by comparing the cone to a pyramid and applying
Cavalieri's principle In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...
– specifically, comparing the cone to a (vertically scaled) right square pyramid, which forms one third of a cube. This formula cannot be proven without using such infinitesimal arguments – unlike the 2-dimensional formulae for polyhedral area, though similar to the area of the circle – and hence admitted less rigorous proofs before the advent of calculus, with the ancient Greeks using the
method of exhaustion The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in are ...
. This is essentially the content of Hilbert's third problem – more precisely, not all polyhedral pyramids are ''scissors congruent'' (can be cut apart into finite pieces and rearranged into the other), and thus volume cannot be computed purely by using a decomposition argument.


Center of mass

The
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
of a conic solid of uniform density lies one-quarter of the way from the center of the base to the vertex, on the straight line joining the two.


Right circular cone


Volume

For a circular cone with radius ''r'' and height ''h'', the base is a circle of area \pi r^2 and so the formula for volume becomes :V = \frac \pi r^2 h.


Slant height

The slant height of a right circular cone is the distance from any point on the
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
of its base to the apex via a line segment along the surface of the cone. It is given by \sqrt, where r is the
radius In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...
of the base and h is the height. This can be proved by the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
.


Surface area

The lateral surface area of a right circular cone is LSA = \pi r l where r is the radius of the circle at the bottom of the cone and l is the slant height of the cone. The surface area of the bottom circle of a cone is the same as for any circle, \pi r^2. Thus, the total surface area of a right circular cone can be expressed as each of the following: *Radius and height :\pi r^2+\pi r \sqrt :(the area of the base plus the area of the lateral surface; the term \sqrt is the slant height) :\pi r \left(r + \sqrt\right) :where r is the radius and h is the height. *Radius and slant height :\pi r^2+\pi r l :\pi r(r+l) :where r is the radius and l is the slant height. *Circumference and slant height :\frac + \frac 2 :\left(\frac c 2\right)\left(\frac c + l\right) :where c is the circumference and l is the slant height. *Apex angle and height :\pi h^2 \tan \frac \left(\tan \frac + \sec \frac\right) :where \theta is the apex angle and h is the height.


Circular sector

The
circular sector A circular sector, also known as circle sector or disk sector (symbol: ⌔), is the portion of a disk (a closed region bounded by a circle) enclosed by two radii and an arc, where the smaller area is known as the ''minor sector'' and the large ...
obtained by unfolding the surface of one nappe of the cone has: *radius ''R'' :R = \sqrt *arc length ''L'' :L = c = 2\pi r *central angle ''φ'' in radians :\phi = \frac = \frac


Equation form

The surface of a cone can be parameterized as :f(\theta,h) = (h \cos\theta, h \sin\theta, h ), where \theta \in [0,2\pi) is the angle "around" the cone, and h \in \mathbb is the "height" along the cone. A right solid circular cone with height h and aperture 2\theta, whose axis is the z coordinate axis and whose apex is the origin, is described parametrically as :F(s,t,u) = \left(u \tan s \cos t, u \tan s \sin t, u \right) where s,t,u range over [0,\theta), [0,2\pi), and [0,h], respectively. In Implicit function, implicit form, the same solid is defined by the inequalities :\, where :F(x,y,z) = (x^2 + y^2)(\cos\theta)^2 - z^2 (\sin \theta)^2.\, More generally, a right circular cone with vertex at the origin, axis parallel to the vector d, and aperture 2\theta, is given by the implicit
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
equation F(u) = 0 where :F(u) = (u \cdot d)^2 - (d \cdot d) (u \cdot u) (\cos \theta)^2   or   F(u) = u \cdot d - , d, , u, \cos \theta where u=(x,y,z), and u \cdot d denotes the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
.


Elliptic cone

In the
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, an ''elliptic cone'' is the locus of an equation of the form : \frac + \frac = z^2 . It is an affine image of the right-circular ''unit cone'' with equation x^2+y^2=z^2\ . From the fact, that the affine image of a
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
is a conic section of the same type (ellipse, parabola,...) one gets: *Any ''plane section'' of an elliptic cone is a conic section. Obviously, any right circular cone contains circles. This is also true, but less obvious, in the general case (see
circular section In geometry, a circular section is a circle on a quadric surface (such as an ellipsoid or hyperboloid). It is a special plane section of the quadric, as this circle is the intersection with the quadric of the plane containing the circle. Any plan ...
). The intersection of an elliptic cone with a concentric sphere is a spherical conic.


Projective geometry

In
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
, a
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an ...
is simply a cone whose apex is at infinity. Intuitively, if one keeps the base fixed and takes the limit as the apex goes to infinity, one obtains a cylinder, the angle of the side increasing as
arctan In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spe ...
, in the limit forming a
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. Th ...
. This is useful in the definition of degenerate conics, which require considering the cylindrical conics. According to
G. B. Halsted George Bruce Halsted (November 25, 1853 – March 16, 1922), usually cited as G. B. Halsted, was an American mathematician who explored foundations of geometry and introduced non-Euclidean geometry into the United States through his own work and ...
, a cone is generated similarly to a Steiner conic only with a projectivity and axial pencils (not in perspective) rather than the projective ranges used for the Steiner conic: "If two copunctual non-costraight axial pencils are projective but not perspective, the meets of correlated planes form a 'conic surface of the second order', or 'cone'."
G. B. Halsted George Bruce Halsted (November 25, 1853 – March 16, 1922), usually cited as G. B. Halsted, was an American mathematician who explored foundations of geometry and introduced non-Euclidean geometry into the United States through his own work and ...
(1906) ''Synthetic Projective Geometry'', page 20


Generalizations

The definition of a cone may be extended to higher dimensions; see convex cone. In this case, one says that a
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
''C'' in the real
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
R''n'' is a cone (with apex at the origin) if for every vector ''x'' in ''C'' and every nonnegative real number ''a'', the vector ''ax'' is in ''C''. In this context, the analogues of circular cones are not usually special; in fact one is often interested in '' polyhedral cones''. An even more general concept is the topological cone, which is defined in arbitrary topological spaces.


See also

* Bicone *
Cone (linear algebra) In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every . W ...
*
Cylinder (geometry) A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infi ...
*
Democritus Democritus (; el, Δημόκριτος, ''Dēmókritos'', meaning "chosen of the people"; – ) was an Ancient Greek pre-Socratic philosopher from Abdera, primarily remembered today for his formulation of an atomic theory of the universe. No ...
* Generalized conic *
Hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
*
List of shapes Lists of shapes cover different types of geometric shape and related topics. They include mathematics topics and other lists of shapes, such as shapes used by drawing or teaching tools. Mathematics * List of mathematical shapes * List of two- ...
*
Pyrometric cone Pyrometric cones are pyrometric devices that are used to gauge heatwork during the firing of ceramic materials. The cones, often used in sets of three, are positioned in a kiln with the wares to be fired and provide a visual indication of when ...
* Quadric *
Rotation of axes In mathematics, a rotation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x′y′''-Cartesian coordinate system in which the origin is kept fixed and the ''x′'' and ''y′'' axes are ...
* Ruled surface *
Translation of axes In mathematics, a translation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x'y-Cartesian coordinate system in which the ''x axis is parallel to the ''x'' axis and ''k'' units away, and the ''y ...


Notes


References

*


External links

* * * {{MathWorld , urlname=GeneralizedCone , title=Generalized Cone * An interactiv
Spinning Cone
from Maths Is Fun
Paper model cone



Cut a Cone
An interactive demonstration of the intersection of a cone with a plane Elementary shapes Surfaces